47. chen.ruau.ea:26:bifurcation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.. centered:: Chemotaxis models with signal-dependent sensitivity and a logistic-type source, III: Bifurcation and spike-like solutions
.. centered:: Le Chen, Ian Ruau, and Wenxian Shen

**Abstract:** This third paper in a series on a parabolic-elliptic chemotaxis
model studies bifurcation of positive constant equilibria in one space
dimension. We consider

.. math::

   \begin{cases}
   u_t=\Delta u-\chi_0\nabla\cdot\left(\dfrac{u^m}{(1+v)^\beta}\nabla v\right)+au-bu^{1+\alpha}, & x\in\Omega,\\
   0=\Delta v-\mu v+\nu u^\gamma, & x\in\Omega,\\
   \dfrac{\partial u}{\partial n}=\dfrac{\partial v}{\partial n}=0, & x\in\partial\Omega,
   \end{cases}

where :math:`\Omega` is a bounded interval, :math:`\alpha,\gamma,m,\mu,\nu>0`,
:math:`\chi_0\in\mathbb{R}`, and :math:`a,b,\beta\ge 0`. Building on earlier
results on global existence, boundedness, persistence, and stabilization, we
take :math:`\chi_0` as the bifurcation parameter and examine the effects of
:math:`\beta` and the domain size on the bifurcation.

Center manifold reduction yields a coefficient :math:`\beta_{n_0}` at a simple
bifurcation mode :math:`n_0` of a positive constant equilibrium. The pitchfork
bifurcation is supercritical if :math:`\beta_{n_0}>0` and subcritical if
:math:`\beta_{n_0}<0`. Stronger desensitization raises the sensitivity required
for loss of linear stability. Bifurcation may be supercritical for small
:math:`\beta` but is always subcritical for sufficiently large :math:`\beta`.
Thus, desensitization makes patterns harder to trigger and favors abrupt rather
than continuous emergence. The bifurcation mode increases with domain size,
allowing more complex spatial patterns.

Under suitable conditions, we prove unbounded continuation of local bifurcation
branches. We also show that large-sensitivity stationary sequences satisfying a
peak-gap hypothesis develop spike-like profiles. Numerical continuation and
time-evolution experiments illustrate the bifurcating branches and spike-like
solutions and suggest unbounded continuation and increasingly sharp stationary
profiles.

**Keywords:** chemotaxis; signal-dependent sensitivity; logistic source;
pitchfork bifurcation; center manifold reduction; global continuation;
spike-like solutions. **MSC:** 35B32, 35K45, 35M31, 35Q92, 92C17, 92D25.

`Preprint <https://arxiv.org/abs/2610.20976>`_

:cite:`chen.ruau.ea:26:bifurcation` Le Chen, Ian Ruau & Wenxian Shen (2026) 'Chemotaxis models with signal-dependent sensitivity and a logistic-type source, III: Bifurcation and spike-like solutions', *preprint arXiv:2610.20976, 42 pages*

.. code-block:: bibtex

   @article{chen.ruau.ea:26:bifurcation,
      title         = {Chemotaxis models with signal-dependent sensitivity and a logistic-type source, {III}: Bifurcation and spike-like solutions},
      author        = {Le Chen and Ian Ruau and Wenxian Shen},
      year          = {2026},
      month         = {October},
      journal       = {Preprint arXiv:2610.20976},
      url           = {http://arXiv.org/abs/2610.20976}
   }


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